3.139 \(\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx\)

Optimal. Leaf size=530 \[ \frac{3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac{(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac{3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac{h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+4;-n,-p;m+5;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]

[Out]

((b*g - a*h)^3*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[1 + m, -n, -p, 2 + m, -((d*(a + b*x))/(b*c -
a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e - a*f))^
p) + (3*h*(b*g - a*h)^2*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[2 + m, -n, -p, 3 + m, -((d*(a + b*x
))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e
- a*f))^p) + (3*h^2*(b*g - a*h)*(a + b*x)^(3 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[3 + m, -n, -p, 4 + m, -((d
*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(3 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f
*x))/(b*e - a*f))^p) + (h^3*(a + b*x)^(4 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[4 + m, -n, -p, 5 + m, -((d*(a +
b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(4 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/
(b*e - a*f))^p)

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Rubi [A]  time = 1.17864, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {181, 159, 140, 139, 138} \[ \frac{3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+3;-n,-p;m+4;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac{(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac{3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+2;-n,-p;m+3;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac{h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+4;-n,-p;m+5;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3,x]

[Out]

((b*g - a*h)^3*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[1 + m, -n, -p, 2 + m, -((d*(a + b*x))/(b*c -
a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e - a*f))^
p) + (3*h*(b*g - a*h)^2*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[2 + m, -n, -p, 3 + m, -((d*(a + b*x
))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e
- a*f))^p) + (3*h^2*(b*g - a*h)*(a + b*x)^(3 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[3 + m, -n, -p, 4 + m, -((d
*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(3 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f
*x))/(b*e - a*f))^p) + (h^3*(a + b*x)^(4 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[4 + m, -n, -p, 5 + m, -((d*(a +
b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(4 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/
(b*e - a*f))^p)

Rule 181

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] + Dist[(b*g - a*
h)/b, Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n
, p}, x] && IGtQ[q, 0] && (SumSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(
c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || ( !SumS
implerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 140

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c
- a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 139

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rubi steps

\begin{align*} \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx &=\frac{h \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}+\frac{(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}\\ &=\frac{h^2 \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+2 \frac{(h (b g-a h)) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+\frac{(b g-a h)^2 \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}\\ &=\frac{h^3 \int (a+b x)^{3+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac{\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+2 \left (\frac{\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}\right )+\frac{(b g-a h)^3 \int (a+b x)^m (c+d x)^n (e+f x)^p \, dx}{b^3}\\ &=\frac{\left (h^3 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{3+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac{\left (h^2 (b g-a h) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+2 \left (\frac{\left (h^2 (b g-a h) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}\right )+\frac{\left ((b g-a h)^3 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}\\ &=\frac{\left (h^3 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{3+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac{\left (h^2 (b g-a h) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{2+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}+2 \left (\frac{\left (h^2 (b g-a h) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{2+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}+\frac{\left (h (b g-a h)^2 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}\right )+\frac{\left ((b g-a h)^3 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{b^3}\\ &=\frac{(b g-a h)^3 (a+b x)^{1+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (1+m)}+\frac{h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (2+m;-n,-p;3+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac{h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (3+m;-n,-p;4+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}+2 \left (\frac{h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (2+m;-n,-p;3+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac{h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (3+m;-n,-p;4+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}\right )+\frac{h^3 (a+b x)^{4+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (4+m;-n,-p;5+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^4 (4+m)}\\ \end{align*}

Mathematica [F]  time = 4.52404, size = 0, normalized size = 0. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3,x]

[Out]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3, x]

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Maple [F]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p} \left ( hx+g \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (h x + g\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="maxima")

[Out]

integrate((h*x + g)^3*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="fricas")

[Out]

integral((h^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3)*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**p*(h*x+g)**3,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="giac")

[Out]

Timed out